34 Mr. 0. Heaviside on Duplex Telegraphy. 



current each station receives from the other through its recei- 

 ving instrument g, and let both A and B send the same cur- 

 rent to line. Then, from the identity of the arrangement at 

 each end, there will be no current in the line, which may be 

 removed without influencing the currents in the other conduc- 

 tors. Thus we find 



p_ E a 



" t\ r i a ( h + v) a + b+g 

 or 



Q=S V+<W+g) + a(f+c + b+g)' * ' (1) 



Now this expression for the strength of the received current 

 contains the constants E,/, and g y and the variables a, b, and 

 c. The last two are independent ; but the first is a function of 



all the resistances : for a= — , and 



x 



. <jb.v(c +/) + efxjg + b) + bc(g + c)(b +/) -, 



' " (ff + b)(c+/)x + bc(ff + b + c+f) ■ ■ W 



This gives a quadratic equation for the determination of x, 



which, however, it is unnecessary to effect. By eliminating 



be 

 «= -7 from (1), we have 



G= <j+c)(b+c,).v+bc(f+c+b+ g y • ' 



We have to make Gr a maximum with respect to b and c ; and 



therefore we must have -vr = 

 db 



the following conditions : — 



therefore we must have -^-=0 and -^- =0. Thus we have 

 db dc 



ff z(f+c)-b*c=b(f+cXb+y)^ 



Mb+g)-b<?=c(f+c)(b + gf£. 



The only difficulty now lies with the complex function x. 



It would be most natural to obtain x. C -£j and -=~ as functions 



db dc 



°f h c >f> 9i an d l> aj id then find the values of b and c, in terms 



of the constants /, g, and I, which make G a maximum. But 



it will be found impossible to obtain an explicit solution in this 



manner, owing to the high degree of the final equations. How- 



